To sum up, when the expression contains either sums or products of the (secret) codes, the problem of showing publicly the numerical value of this expression is trivially solved. The issue arises when the expression is composed by a combination of sums and products of codes. The aim of this work is to present a protocol able to show publicly addition and multiplication on secretly shared values. This problem has already been solved by Donald Beaver in 1991 by using traditional algebra - essentially based on the number theory - and a wide series of circuits garbling [1]}. Unfortunately, Beaver's protocol require many rounds of communication and this slows down the machine time with consequent expensive energy. To overcome these obstacles we resort to a more sophisticated mathematical approach, based on the use of masks and FMPC (Fourier-based Multi-Party Computation) [2]}. Our protocol operates by using the masks’ method and paves the way for a new kind of multiparty computation protocols capable of computing sums and multiplications of secrets as an alternative to circuit garbling. More precisely, we hide the users' input codes within the cosine components of the Fourier series of the main function (denoted by \(f(x)\) ), combined with additive masks that can be chosen arbitrarily by the users. The result of the function is visible publicly but, of course, the user's mask is kept secret. Users send the secret codes hidden by the masks to four nodes. Computations are performed by using a new algebra, referred to as the \(\Theta ^{[n]}\) -algebra (or Theta-algebra). Finally, the results of the calculation are transmitted to the display which, thanks to the generalised Parseval's identity for Fourier series applicable to an arbitrary number of inputs, can show publicly the numerical result of the mathematical expression while keeping secret the users' codes. The participants' privacy is guaranteed if at least three nodes are not corrupted. Successively, we solved the problem using \(3f+1\)  nodes, grouped in four categories. In this case users, besides the additive mask, arbitrarily choose another multiplicative mask. Even for multiple nodes, the participants' privacy is guaranteed if at least three nodes belonging to three different categories are not corrupted and at least one node, belonging to the second level of computation, is not corrupted. Compared to the original Beaver's method, our protocol is capable to compute addition and multiplication with no online communication and its simplicity provides efficiency and ease of implementation.
